# Recent MathOverflow Questions

### Numerical range of a pair of operators

Math Overflow Recent Questions - Fri, 04/20/2018 - 13:26

Let $M\in \mathcal{B}(F)^+$ and $S_1,S_2\in \mathcal{B}(F)$ where $F$ is an infinite-dimensional complex Hilbert space. We consider $$W(S_1,S_2)=\{(\langle MS_1x, x\rangle,\langle MS_2x, x\rangle);\;\;x\in F,\;\langle Mx,x \rangle=1\}.$$

Assume that $S_1$ and $S_2$ are $M$-self adjoint (i.e. $MS_1=S_1^*M$ and $MS_2=S_2^*M$) such that $S_1S_2=S_2S_1$ and $MS_k=S_kM,\,k=1,2.$ It is possible to show that $W(S_1,S_2)$ is convex?

Since $S_1$ and $S_2$ are commuting and $M$-self adjoint such that $MS_k=S_kM,\,k=1,2$, then $\exists\,(X,\mu)$, $\varphi_1,\varphi_2\in L^\infty(\mu)$ and a unitary operator $U:F\longrightarrow L^2(\mu)$, such that $$U(MS_k)U^*h=\varphi_kh,\;\forall h\in L^2(\mu),\,k=1,2.$$

Let $\lambda=(\lambda_1,\lambda_2)$, $\eta=(\eta_1,\eta_2)$ be any pair of point in $W(S_1,S_2)$, then there exist $f,g \in L^2(X, \mu)$ such that for all $k= 1,2$ we have $$\lambda_k=\displaystyle\int_X \varphi_k|f|^2d\mu\;\;\text{and}\;\;\eta_k=\displaystyle\int_X \varphi_k|g|^2d\mu.$$ Let $\xi=t\lambda+(1-t)\eta,\;t\in[0,1]$. So $$\xi_k=t\displaystyle\int_X \varphi_k|f|^2d\mu+(1-t)\displaystyle\int_X \varphi_k|g|^2d\mu.$$

It is possible to show that $\xi\in W(S_1,S_2)$?

### A cohomology associated to a Riemannian manifold

Math Overflow Recent Questions - Fri, 04/20/2018 - 12:59

Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(X),\; g\in G, X\in M$, the push forward of $X$ under the isometry $g$. So $M$ is a $G$ module.

How can one express the group cohomologies $H^n(G,M)$ explicitely?Is there a reference which contain such computations? What can be said about a riemannian manifold whose all cohomology groups $H^n(G,M)$ vanish?

Edit: according to the comment of Neal I understand the following part of the previous version is a trivial question:

Does this sequence of cohomologies determine the geometry of $N$? Namely is it true to say that two nonisometric metrics on $N$ give two different cohomology sequence?

### Counting primitive lattice points

Math Overflow Recent Questions - Fri, 04/20/2018 - 07:49

In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result):

Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then $$\# \{ (x_1,x_2) \in \Lambda: \max_i |x_i| \leq B, \gcd(x_1,x_2) = 1\} \leq 16\left (\frac{B^2}{d(\Lambda)} + 1\right).$$

My question is whether this generalises to arbitrary dimensions.

Does an analogous result hold for lattices in $\Lambda \subset \mathbb{Z}^n$? Namely, is $$\# \{ (x_1,\dots, x_n) \in \Lambda: \max_i |x_i| \leq B, \gcd(x_1,\dots,x_n) = 1\} \leq C_n\left (\frac{B^n}{d(\Lambda)} + 1 \right)$$ for some constant $C_n$?

If it helps, I'm primarily interested in the case $n=3$.

Obviously I'm aware of standard lattice point counting techniques, but these usually give an error term of the shape $O(\text{boundary of the region/first successive minima})$, and I don't know how to control this in my case. So I'm just looking for uniform upper bounds where this term doesn't appear.

[1] Heath-Brown - Diophantine approximation with Square-free numbers

### Homology of $\mathrm{PGL}_2(F)$

Math Overflow Recent Questions - Fri, 04/20/2018 - 03:58

Here is a hopelessly naive question. Please point me to the relevant literature!

Let $F$ be any field. The Cartan subgroup of $\mathrm{PGL}_2(F)$ is $F^\times\rtimes \Sigma_2$. Let $X_2(F)$ be the homotopy cofiber of $B(F^\times\rtimes\Sigma_2)\to B\mathrm{PGL}_2(F)$, so there is a long exact sequence $$\ldots\to H_i(F^\times\rtimes\Sigma_2)\to H_i(\mathrm{PGL}_2(F))\to H_i(X_2(F))\to \ldots$$ (I'm not sure if I should write $H_i(G)$ or $H_i(BG)$...). If I am not making a stupid mistake, then the known computations of the homology of $\mathrm{GL}_2(F)$ going up to the first unstable group $H_3(\mathrm{GL}_2(F))$ can be summarized by saying that $H_i(X_2(F))=0$ for $i=1,2$, and $H_3(X_2(F))=\mathfrak p(F)$ is the pre-Bloch group of $F$. (The pre-Bloch group is the quotient of $\mathbb Z[F^\times]$ by the $5$-term relations''.) (I might be off by some $2$-torsion.) By Hurewicz (and the check that $\pi_1 X_2(F)=0$), the same holds true for the homotopy groups of $X_2(F)$.

As far as I could find, little is known about the homology of $\mathrm{GL}_2(F)$ or $\mathrm{PGL}_2(F)$ in degrees $>3$. The rational structure could be determined by the following very naive question.

Question. Are the homotopy groups $\pi_i X_2(F)$ bounded torsion for $i\neq 3$?

I might even expect the implicit bound to be independent of $F$ (but of course depend on $i$; the order of magnitude should be $i!$).

The only evidence I have is that a back-of-the-envelope calculation seems to suggest that this holds true for finite fields. In that case the order of $\mathrm{PGL}_2(\mathbb F_q)$ is the product of $q-1$, $q$ and $q+1$. Localized at $q-1$, the homology agrees with the homology of the Cartan; localized at $q$, it is only in degrees larger than $q$ (which is OK as the bound on torsion may depend on $i$), and localized at $q+1$, the homology seems to agree with the homology of a $K(\mathbb Z/(q+1),3)$ (up to bounded torsion in each degree), which is roughly $\mathbb Z/(q+1)$ in all degrees $\equiv 3\mod 4$ (and is bounded torsion in other degrees). The pre-Bloch group $\mathfrak p(\mathbb F_q)$ is also $\mathbb Z/(q+1)$ up to $2$-torsion, so things add up.

The case of $F=\mathbb Q$ might be amenable to computations, but I was unable to do those. Are there any relevant results about $H_4(\mathrm{GL}_2(F))$ that might shed light on $\pi_4 X_2(F)$?

A final remark: I expect that it is critical that $F$ is a field. The similar statement should be false already for discrete valuation rings, or $\mathbb Z[\tfrac 1n]$ (which is why the case of $F=\mathbb Q$ is not so easy to compute; I presume one would try to first compute for all $\mathbb Z[\tfrac 1n]$ and then pass to a colimit, and the desired structure should only appear in the colimit).

### Infinite-time Turing machines and the formal Church's thesis

Math Overflow Recent Questions - Thu, 04/19/2018 - 12:16

Infinite-time Turing machines are known to either halt or loop in countable time.

In the spirit of double-negation translation, is there a statement which is: classically equivalent to this; consistent with the formal Church's thesis, that all total functions on the integers are recursive; and (this part is of course subjective) which actually tells us something "meaningful or informative" beyond the vacuous sense in which any such translation does?

### Stabilizer of two short exact sequences at the same time

Math Overflow Recent Questions - Thu, 04/19/2018 - 07:51

For two short exact sequences of say, finitely generated modules of some ring, $0\rightarrow N\xrightarrow{a} R\xrightarrow{b} M\rightarrow0, 0\rightarrow K\xrightarrow{a'}R\xrightarrow{b'}L\rightarrow0.$ Let $\phi \in Aut(R)$ and $\phi$ acting on these two short exact sequences by $\phi.a=\phi\circ a, \phi.b=b\circ\phi^{-1},\phi.a'=\phi\circ a', \phi.b'=b'\circ\phi^{-1}.$ My question is that the book I am reading says that by some diagram chasing, the cardinality of stabilizer of this action of $Aut(R)$ is the same as the cardinality of $Hom(Coker\space b'a, Ker\space b'a).$ I have no idea how to do explain it, can somebody help me?

### What, mathematically speaking, does it mean to say that the continuation monad can simulate all monads?

Math Overflow Recent Questions - Thu, 04/19/2018 - 04:47

In various places it is stated that the continuation monad can simulate all monads in some sense (see for example http://lambda1.jimpryor.net/manipulating_trees_with_monads/)) In particular, in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.8213&rep=rep1&type=pdf it is claimed that

any monad whose unit and extension operations are expressible as purely functional terms can be embedded in a call-by-value language with “composable continuations”.

I was wondering what (Category-theoretic) mathematical content these claims of simulation have, and what precisely they show us about monads and categories in mathematical terms. In what ways is the continuation monad special, mathematically, compared to other monads, if at all? (I seem to remember some connection between the Yoneda embedding and continuations which might be relevant https://golem.ph.utexas.edu/category/2008/01/the_continuation_passing_trans.html, although I don't know)

One other relevant fact might be that the continuation monad is the monad which takes individuals $\alpha$ to the principal ultrafilters containing them (that is it provides the map $\alpha \mapsto (\alpha \rightarrow \omega) \rightarrow \omega$) )

Edit: I have been asked to explain what I mean by the continuation monad. Suppose we have a monad mapping types of the simply typed lambda calculus to types of the simply typed lambda calculus (the relevant types of this calculus are of two kinds: (1) the basic types, (I.e, the type $e$ of individuals type belonging to domain $D_e$ and the type of truth values belonging to domain $D_t$) and, (2) for all basic types $\alpha, \beta$, the type of functions between objects of type $\alpha$ and objects of type $\beta$ belonging to domain $D_{\beta}^{D_{\alpha}}$). Let $\alpha, \beta$ denote types and $\rightarrow$ a mapping between types. Let $a : \alpha \hspace{0.2cm}$ (or $b: \beta)$ indicate that $a\hspace{0.2cm}$ (or $b$) is an expression of type $\alpha \hspace{0.2cm}$ (or $\beta)$. Let $\lambda x. t$ denote a function from objects of the type of the variable $x$, to objects of the type of $t$, as in the simply typed lambda calculus. Then a continuation monad is a structure $\thinspace(\mathbb{M}, \eta, ⋆)\thinspace$, with $\mathbb{M}$ an endofunctor on the category of types of the simply typed lambda calculus, $\eta$ the unit (a natural transformation) and ⋆ the binary operation of the monoid) such that:

$$\mathbb{M} \thinspace α = (α → ω) → ω, \hspace{1cm} ∀α$$ $$η(a) = λc. c(a) : \mathbb{M} \thinspace α \hspace{1cm} ∀a : α$$ $$m ⋆ k = λc. m (λa. k(a)(c)): \mathbb{M}\thinspace β \hspace{1cm} ∀m : \mathbb{M}\thinspace α, k : α → \mathbb{M}\thinspace β .$$

The continuation of an expression $a$ is $\eta(a) = (a \rightarrow \omega) \rightarrow \omega$.

Edit 2: let $\omega$ denote some fixed type, such as the type of truth values (i.e, $\{ \top, \bot\}$)

### Bredon cohomology of $\mathbb{S}^\sigma$

Math Overflow Recent Questions - Wed, 04/18/2018 - 12:17

I tried to compute Bredon cohomology of $\mathbb{S}^\sigma$, where $\sigma$ is a sign representation of $\mathbb{Z}/2$, following first chapter and first construction of cohomology from Bredon's "Equivariant cohomology theories". Could somebody please verify it, at least the result?

Throughout $G$ means $\mathbb{Z}/2$.

So I assume the following: $G$-CW structure is obvious, given by two points with trivial action as 0-cells and two arcs with swapping action as 1-cells. I am using simple coefficient system $\mathcal{L}$ on it, that is my functor from "cellular category" to abelian groups factors through some coefficient system $M$. $M$ consists of two groups $M(*)$ - trivial $G$-module and $M(G)$ - $G$-module, and an equivariant map $\epsilon:M(*)\rightarrow M(G)$.

$C^0(\mathbb{S}^\sigma;\mathcal{L})$ consists of the functions $f:\{e^0_1,e^0_2\}\rightarrow M(*)$. Since action of G is trivial on 0-cells, induced action on 0-chains is also trivial, therefore $C^0(\mathbb{S}^\sigma;\mathcal{L})=C^0_G(\mathbb{S}^\sigma;\mathcal{L})=M(*)^2$. $C^1(\mathbb{S}^\sigma;\mathcal{L})$ consists of the functions $f:\{e^1_1,e^1_2\}\rightarrow M(G)$. Action on 1-cells is non-trivial (even free), so $C^1_G(\mathbb{S}^\sigma;\mathcal{L})$ consists of equivariant $f$'s. Thus $C^1_G(\mathbb{S}^\sigma;\mathcal{L})=M(1)$.

The only non-trivial differential is $\delta :C^0\rightarrow C^1$ and is given by $(\delta f)(\tau)=\pm\epsilon(f(e^0_1))\mp\epsilon(f(e^0_2))$. Here $\tau$ of course means any of two 1-dimensional cells.

So $H^0_G(\mathbb{S}^\sigma;\mathcal{L})=M(*)$ and $H^1_G(\mathbb{S}^\sigma;\mathcal{L})=M(G)/M(G)^G$ - but for this I have to assume that $\epsilon$ is an iso on $M(G)^G$.

If this is not "mathoverflow" question, I can ask it also on MathStack.

### Can we deduce the sign of this integral which includes cosine transform?

Math Overflow Recent Questions - Wed, 04/18/2018 - 11:50

Let's $v(t)$ be a negative real function and consider the following integral with $f(t)$ complex (rapidly decreasing at infinity):

$$A= \int\limits_{0}^\infty \frac{1}{x} \int\limits_{x}^\infty \Big(v(t)f'(t) + \frac{1}{2} v'(t)f(t) \Big) \overline{f(t)} + \overline{\Big(v(t)f'(t) + \frac{1}{2} v'(t)f(t) \Big) }f(t) dt dx$$

Providing the integral is well defined, it is not hard to see that $A$ is always positive ($v(t)$ is negative by hypothesis) by an integration by parts we directly have:

$$A= \int\limits_{0}^\infty \frac{1}{x} [v(t) f(t) \overline{f(t)} ]_{x}^\infty dx = - \int\limits_{0}^\infty \frac{1}{x} v(x) f(x) \overline{f(t)} dx$$

Now, can we say something about the sign of the same expression where we consider the cosine transform of precedent functions ? (we note $\mathcal{F_c}$ the cosine transform)

$$B=\int\limits_{0}^\infty \frac{1}{x} \int\limits_{x}^\infty \mathcal{F_c}\Big(v(t)f'(t) + \frac{1}{2} v'(t)f(t) \Big) \overline{\mathcal{F_c}(f(t))} + \overline{\mathcal{F_c}\Big(v(t)f'(t) + \frac{1}{2} v'(t)f(t) \Big) } \mathcal{F_c}(f(t)) dt dx$$

I do not manage to use Parseval's equality. May be we cannot say anything about the sign of $B$ ? Any idea on how to investigate this sign ? Or conditions on $v$ and $f$ to have it also positive ?

(Note that to have integral $A$ converging we need : $$\int\limits_{0}^\infty \Big(v(t)f'(t) + \frac{1}{2} v'(t)f(t) \Big) \overline{f(t)} + \overline{\Big(v(t)f'(t) + \frac{1}{2} v'(t)f(t) \Big) }f(t) dt =0$$)

### How does raising the zeta function to 1/lnx transform the properties of the zeta function? [on hold]

Math Overflow Recent Questions - Wed, 04/18/2018 - 11:43

How does pre-exponentiating the zeta function by 1/lnx changes the function in terms of its properties. How are the zeros changed, if at all?

### The stalk of the sheaf of sheaves

Math Overflow Recent Questions - Wed, 04/18/2018 - 11:40

For $X$ a reasonable topological space denote by $Sh(X)$ the derived (stable $\infty$-) category of sheaves of vector spaces (over a fixed field $k$) on $X$.

Consider the filtered diagram whose whose objects are $Sh(U)$ for all $0 \in U \subset \mathbb{R}^n$ open subsets. And morphisms are restriction functors $j^*:Sh(V) \to Sh(U)$ for every inclusion $j : U \to V$.

Question: What is the ($\infty$-)colimit of this diagram?

In slightly less precise terms i'm looking for the stalk at $0$ of the sheaf of categories on $\mathbb{R}^n$ which assigns to any open set the category of sheaves on that open.

I have a hunch that this colimit is equivalent to the category of $\mathbb{R}_{>0}$-equivariant sheaves on $\mathbb{R}^n$ but I have no idea how to prove this...

### Optimization with bounds on the control and its derivative

Math Overflow Recent Questions - Wed, 04/18/2018 - 11:26

I would like to understand the following optimization problem. Let $F(t,x)$ be a continuous function defined on $[0,1]\times [0,1]$, which is increasing in $t$ and convex in $x$ (I have in mind $F(t,x)=x(t-\frac{x}{2})$), the goal is to solve: \begin{align*} \min_{\alpha \in \mathcal{C}_A} \int_0^1F(t,x_t)dt \end{align*} subject to : \begin{align*} x_t \in \arg \max_{x \in [0,1]} F(t,x)-\alpha(x)1\!\!1_{x\neq 0} \end{align*} and $\mathcal{C}_A$ is the class of differentiable functions such that: \begin{align*} \int_0^1\alpha(x_t)dt &\leq A,\\ \forall x\in [0,1], 0 \leq \alpha(x) & \leq K, \text{and }\vert \alpha'(x)\vert \leq K, \end{align*} where $K> 0$ is a fixed constant.

What annoys me the most is the boundness conditions on the function $\alpha$ and its derivative. Would you have any hint or reference which would help me to get started ?

### Are there any non-asymptotic bounds for the minimum empirical risk vs theoretical risk?

Math Overflow Recent Questions - Wed, 04/18/2018 - 10:13

I'm trying to see if there's any bounds on the difference between $f_{ERM}$ and $f^{*}$. For now, define $\mathcal{F}$ to be a function class.

Let $P$ be a probability measure and $\hat{P_n}$ be the empirical measure on a set $S\subset \mathbb{R}^2$.

Let $$f^{*} = \arg \min_{f\in \mathcal{F}} R_P(f)$$ and

$$\hat{f^{*}} = \arg \min_{f\in \mathcal{F}} R_{\hat{P_n}}(f)$$ where $R_P$ denotes the classification risk with respect to the measure $P$ (and similarly for $\hat{P_n}$). So $R_P(f)=E_{(x,y)\sim P}[L(f(x),y)]$ and $R_\hat{P_n}(f)=\frac{1}{n}\sum_{i=1}^n L(f(x_i),y_i)$ where $x_i,y_i$ are iid samples drawn from distribution $P$.

Is there anything we know about $\| R_{\hat{P_n}} (\hat{f^{*}}) - R_{P}(f^{*}) \|$?

I thought this was related to ERM stability bounds except those bounds are for $\| R_{P} (\hat{f^{*}}) - R_{P}(f^{*}) \|$

### Is there a name for sequences of integers reduced to their lowest prime divisors?

Math Overflow Recent Questions - Wed, 04/18/2018 - 09:46

When trying to obtain the value of Jacobsthal's function for some $n$; to find the largest sequence of consecutive numbers that are all coprime to $n$, one approach (and the only direct approach that I know of) is to exhaust all of the possible sequences of consecutive numbers that are all coprime to $n$ by trial. But when doing this, it's helpful (when working by hand) to represent all integers (except $1, 0$ and $-1$) by their lowest prime divisor. For example; the sequence $2,3,4,5,6,7,8,9,10$ would be equivalent to $2,3,2,5,2,7,2,3,2$. This representation has it's advantages as it represents more than one sequence of consecutive integers. For example; $2,3,2,5,2,7,2,3,2$ is also the respective representation of $212,213,214,215,216,217,218,219,220$. This representation enables the exhaustive approach for determining the value of Jacobsthal's function for some $n$.

But the thing is, I can't find a specific name for this representation of integers, (or this type of sequence in a more context-free setting). Is there a referable/favoured name for this representation? Usually, I have to make up my own dummy name for them to be able to say anything about them, which I would like to avoid if possible.

For example; I would like to conjecture that for any unknown name of the first $n$ prime numbers, that is larger than $2p_{n-1} -1$, there exist a reflection point in the unknown name of some prime numbers, whom when multiplied together are larger than ${p_n}^2$. This is just another way of conjecturing that a prime number exist in all intervals of length $2p_{n-1}$, bounded above by ${p_n}^2$. That's an open question, but I would like to read more about this sort of approach, and related attempts to identify how these unknown name are distributed in respect to primorial numbers, and squared primes.

### Modular tensor category associated to an even integral lattice and the lattice automorphism

Math Overflow Recent Questions - Wed, 04/18/2018 - 09:44

Let $(L,\langle -,-\rangle)$ be an even integral lattice, and let $(A,q)$ be the associated discriminant form: $$A=L^*/L, \quad q(a)=e^{\pi i \langle a,a\rangle}.$$ This in turn determines a modular tensor category $\mathcal{C}(A,q)$. (This is also the category of modules of the lattice VOA $V_L$ constructed from $L$.)

We then naturally have a group homomorphism $O(L)\to Aut(\mathcal{C}(A,q))$.

So it seems natural to study the obstructions to extending $\mathcal{C}(A,q)$ by $O(L)$ or any of its subgroups in the sense of [ENO, arXiv:0909.3140].

Does anybody know of any references concerning this point? I would be happy with any partial results, or any starting point for me to explore the literature.

(Also, does the extension theory of ENO appear somewhere in the theory of lattice VOA?)

### Neural Network derivative of derivative

Math Overflow Recent Questions - Wed, 04/18/2018 - 04:35

I have a normal Feed Forward NN: $$N\,=\,\sigma(W_2\,(\sigma(W_1x+b_1)+b_2)$$ A penalty term is computed using the derivative of the network w.r.t. the inputs: $L = f(\frac{\partial N}{ \partial x})$, which is, if I am right, $$L = f(\sigma'(out2)*W_2*\sigma'(out1)*W_1)$$ if we call $out_2$ and $out_1$ the outputs of the NN layers before the sigmoid. Now I need to compute the derivative of L w.r.t. the weights $W_2$ and $W_1$, to let my network learn. Since $$\partial L/ \partial W = \frac{\partial L }{ \partial (\partial N/ \partial x)}(\frac {\partial N }{ \partial x}(x, W_1, W_2))*\frac {\partial(\partial N/ \partial x)}{ \partial W}(W)$$where calculating the first term is easy, i need the second terms.

I read these slides at here and found: $\partial (AXB) = A\partial (X)B = BA\partial X$, but as long as I agree with first equality since sigmoid derivatives and the other weight are constants w.r.t. the weight I am searching the derivative for, the second equality doesn't hold for me (and in fact I couldn't demonstrate it) just considering matrix dimensions. Here x = [241x1] (column vector), $W_1$ = [20x241], $W_2$ = [20x20].

### Lagrange Multipliers for two constraints, degenerate case

Math Overflow Recent Questions - Wed, 04/18/2018 - 03:37

To optimize $f(x,y,z)$ subject to $g(x,y,z)=h(x,y,z)=0$, we use the Lagrange Multiplier method and solve \begin{equation*} \nabla f=\lambda \nabla g+\mu\nabla h,\quad g=0,\quad h=0. \end{equation*} Geometrically, $\nabla f$ must lie on the normal plane spanned by $\nabla g$ and $\nabla h$. However, it can happen that $\nabla g$ is parallel to $\nabla h$ at certain points, and hence they cannot span the normal plane. In this case, does $\nabla f$ have to be parallel to $\nabla g$ to be a critical point? If yes, how to explain it? If no, can it happen that some critical points are missing? Thanks.

### Why do we specify the degree of elements in algebra? [migrated]

Math Overflow Recent Questions - Wed, 04/18/2018 - 03:03

In algebra (especially algebraic topology where I have seen it) when we have some sort of ring like $R[x]$ we are quick to specify that $x$ has degree $n$. I understand the importance of this but isn't this just the same as the ring $R[x^n]$?

I would find the later notion more compact and clear given that in early mathematical education $R[x]$ is always assumed to have $x$ in degree 1. In fact then it seems to me that only a single definition of $R[x]$ needs to be given.

I am interested in the historic or good reasons for defining it above. This notion is really common in cohomology for example.

### On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Math Overflow Recent Questions - Wed, 04/18/2018 - 00:33

Let

$$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$

where $\zeta$ denotes the Riemann zeta function.

What are the reasonable asymptotic estimates for $I(T)$ ?

Here is what i think:

It is well-known that

$$\int_{-T}^{T} \log \Big|\zeta\Big(\frac{1}{2} + it\Big)\Big|\mathrm{d}t \ll T\log T$$ Therefore, by a dyadic decomposition, it follows from the above that

$$I(T)=\int_{-T}^{T} \frac{\log |\zeta(\frac{1}{2} + it)|}{\frac{1}{4}+ t^2}\mathrm{d}t \ll\frac{\log T}{T}$$

REMARK: An almost similar application of dyadic decomposition can be found on https://mathoverflow.net/a/285215/123305

### Binary cube root of a given positive, integer square matrix

Math Overflow Recent Questions - Tue, 04/17/2018 - 22:32

How can I find all binary matrices $A$ that are a solution of matrix equation

$$A^3=B$$

for some given positive, integer square matrix $B$? Is there a way to characterize all the solutions?